Energy Stable Discontinuous Galerkin Finite Element Method for the\n Allen-Cahn Equation

2014/09/13 by Bülent Karasözen, Karasözen, Bülent, Ayşe Sarıaydın Filibelioğlu +3
Computer Science · Materials Science · Mathematics · #65L04 #65M60 #65Z05 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.1409.3997

openalex publication_date 2014/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Allen--Cahn equation with constant and degenerate mobility, and with\npolynomial and logarithmic energy functionals is discretized using symmetric\ninterior penalty discontinuous Galerkin (SIPG) finite elements in space. We\nshow that the energy stable average vector field (AVF) method as the time\nintegrator for gradient systems like the Allen-Cahn equation satisfies the\nenergy decreasing property for the fully discrete scheme. The numerical results\nfor one and two dimensional Allen-Cahn equation with periodic boundary\ncondition, using adaptive time stepping, reveal that the discrete energy\ndecreases monotonically, the phase separation and metastability phenomena can\nbe observed and the ripening time is detected correctly.\n

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